Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{2z^2 - 14z + 12}{-2z^3 - 16z^2 + 18z}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {2(z^2 - 7z + 6)} {-2z(z^2 + 8z - 9)} $ $ a = -\dfrac{2}{2z} \cdot \dfrac{z^2 - 7z + 6}{z^2 + 8z - 9} $ Simplify: $ a = - \dfrac{1}{z} \cdot \dfrac{z^2 - 7z + 6}{z^2 + 8z - 9}$ Next factor the numerator and denominator. $ a = - \dfrac{1}{z} \cdot \dfrac{(z - 1)(z - 6)}{(z - 1)(z + 9)}$ Assuming $z \neq 1$ , we can cancel the $z - 1$ $ a = - \dfrac{1}{z} \cdot \dfrac{z - 6}{z + 9}$ Therefore: $ a = \dfrac{ -z + 6 }{ z(z + 9)}$, $z \neq 1$